As seen from QuantumVlasov, the perpendicular momentum only enters in
the equation system through the energy \(\epsilon \) . Consequently, the perpendicular
momentum has a limited effect on the basic physics of the problem, as
pointed out by e.g. Ref [1]} that wrote "It is known from the analysis of the Schwinger effect in spatially homogeneous electric fields that the orthogonal
momentum solely acts as an additional mass term and does
not change the qualitative behavior". Consequently Ref. [1]} put \(p_{\perp }=0\)  in their further analysis. This simplification can be further supported, by plotting the dependence of the pair production
rate on the  \(p_{z}\)  for different perpendicular momenta \(p_{\bot }\) . Considering the number of pairs \(\tilde{n}_\text{loc}(\mathbf {p},t\rightarrow \infty )\)  in Localapprox where we use the configuration of the electric field in Electricfield, the result is
displayed in Figure 1.
We can see that the production rate is diminished
with increasing \(p_{\bot }\) , just as if extra mass has been added to
the electrons and positrons. This indeed confirms the given motivations for
neglecting  the perpendicular momentum in the pair production
process. Particularly if the main aim is just to gain a qualitative
understanding for the dynamics.
